Compression-based asymptotic decomposition: A finite-size scaling framework for entropy rate estimation and data-generating process discrimination

Distinguishing data-generating processes from finite observations is a foundational problem in nonlinear time series analysis. Compression ratios offer a model-free diagnostic of sequential complexity, yet are typically computed at a single sample size, discarding the information contained in how the ratio converges. This paper treats the convergence curve itself as the primary observable: fitting compression ratios across prefix lengths to a three-parameter scaling law, R(N) = a + bN−c, yields three diagnostics from a single analysis. The asymptote a provides a rank-faithful entropy rate proxy (Spearman ρ = 0.977 across 36 processes with known analytical rates), performing on par with the best point-estimate measures. The convergence exponent c captures how quickly the compressor exhausts learnable structure—a dimension not captured by any of the seven benchmark complexity measures tested here, including effort-to-compress (maximum |r| = 0.34). Among generators sharing identical entropy rates, c still varies substantially (mean range 0.173 across 22 same-entropy clusters), distinguishing processes that every point-estimate method considers equivalent. The compression consistency (R2 of the scaling-law fit) serves as a compositional heterogeneity diagnostic: R2 degrades systematically when data from multiple generating processes are temporally concatenated, but not for independent and identically distributed mixtures of the same generators. Deterministic chaotic maps are flagged by Bayesian-information-criterion preference for the logarithmic convergence form motivated by Lempel–Ziv (LZ) asymptotics, manifesting as distinctively low c (<0.40) for generators in the chaotic regime under the power-law parameterization. The framework is validated on 152 synthetic generators spanning 25 families and 105 real-world datasets. Code and data are publicly available at https://github.com/generative-structure/cbad-finite-size-scaling.